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United States

The seat of the U.S. federal government in Washington, D.C. is a federal district known as the "District of Columbia", which is not part of any U.S. state. Other federally administered areas that are within a state, but not under its jurisdiction are called federal enclaves. In main addition, the U.S. government has several other kinds of "federal districts" which are not specifically related to a capital city:

Australia

Russia

There are nine federal districts of Russia, which function as an additional administrative layer between other subdivisions and the Russian Federation as a whole state. But these have nothing to do with the territory surrounding a capital city. However, there are also three cities of federal significance, established by the Constitution — Moscow, Saint Petersburg and Sevastopol. Each city is treated as separate subject of federation, and has its own legislative body. Such status is based on certain special functions which these three cities have, including location of federal government and federal judicial institutions.